O global (strog) defesive alliaces i some product graphs Ismael Gozález Yero (1), Marko Jakovac (), ad Dorota Kuziak (3) (1) Departameto de Matemáticas, Escuela Politécica Superior de Algeciras Uiversidad de Cádiz, Av. Ramó Puyol s/, 110 Algeciras, Spai. ismael.gozalez@uca.es () Faculty of Natural Scieces ad Mathematics Uiversity of Maribor, Koroška 160, 000 Maribor, Sloveia. marko.jakovac@um.si (3) Departamet d Egiyeria Iformàtica i Matemàtiques, Uiversitat Rovira i Virgili, Av. Països Catalas 6, 43007 Tarragoa, Spai. dorota.kuziak@urv.cat May 7, 017 Abstract A defesive alliace i a graph is a set S of vertices with the property that every vertex i S has at most oe more eighbor outside of S tha it has iside of S. A defesive alliace S is called global if it forms a domiatig set. The global defesive alliace umber of a graph G is the miimum cardiality of a global defesive alliace i G. I this article we study the global defesive alliaces i Cartesia product graphs, strog product graphs ad direct product graphs. Specifically we give several bouds for the global defesive alliace umber of these graph products ad express them i terms of the global defesive alliace umbers of the factor graphs. Keywords: Defesive alliaces; global defesive alliaces; domiatio; Cartesia product graphs; strog product graph; direct product graphs. AMS Subject Classificatio Numbers: 05C69; 05C70; 05C76. 1 Itroductio Alliaces i graphs were first described by Kristiase et al. i [10], where alliaces were classified ito defesive, offesive or powerful. After this semial paper, the issue has bee studied itesively. Remarkable examples are the articles [1, 13], where alliaces were geeralized 1
to k-alliaces, ad [6], where the authors preseted the first results o offesive alliaces. To see more iformatio o alliaces i graphs we suggest the recet surveys [18, 19]. Oe of the mai motivatios of this study is based o the NP-completeess of computig the miimum cardiality of (defesive, offesive, powerful) alliaces i graphs. O the other had, several graphs ca be costructed from smaller ad simpler compoets by basic operatios like uios, jois, compositios, or multiplicatios with respect to various products, where properties of the costituets determie the properties of the composite graph. It is therefore desirable to reduce the problem of computig the graphs parameters (alliace umbers, for istace) of product graphs, to the problem of computig some parameters of the factor graphs. Studies o alliaces i product graphs have bee preseted i [1,, 14, 15, 17] where the authors preseted several tight bouds for the (defesive, offesive or powerful) alliace umber of Cartesia product graphs. Also, several exact values for some specific families of Cartesia product graphs were obtaied i these articles. I this sese, we cotiue with these studies for the Cartesia product graphs ad exted them to strog product graphs ad direct product graphs. Sice defesive alliaces defed oly a sigle vertex at a time, Brigham et al. [3] itroduced secure sets which are a geeralizatio of defesive alliaces. Namely, i geeral models, a more efficiet defesive alliace should be able to defed ay attack o the etire alliace or ay part of it. Some geeral results o secure sets were preseted i [4, 5], ad they have also bee studied o differet graph products [3, 9, 16], though exact results are kow oly for a few family of graphs, e.g. grids, cyliders ad toruses. We begi by statig the termiology which will be used. Throughout this article, G = (V,E) deotes a simple graph of order V =. We deote two adjacet vertices u ad v by u v. Give a vertex v V, the set N(v) = {u V : u v} is the eighborhood of v, ad the set N[v] = N(v) {v} is the closed eighborhood of v. So, the degree of a vertex v V is δ(v) = N(v). For a oempty set S V, ad a vertex v V, N S (v) deotes the set of eighbors v has i S, i.e., N S (v) = S N(v). The degree of v i S will be deoted by δ S (v) = N S (v). The complemet of a set S i V is deoted by S. AsetS V isadomiatig set igifforeveryvertexv S, δ S (v) > 0(everyvertexiS is adjacet to at least oe vertex i S). The domiatio umber of G, deoted by γ(g), is the miimum cardiality of a domiatig set i G [8]. A efficiet domiatig set is a domiatig set S = {u 1,u,...,u γ(g) } such that N[u i ] N[u j ] =, for every i,j {1,...,γ(G)}, i j. Examples of graphs havig a efficiet domiatig set are the path graphs P, the cycle graphs C 3k ad the cube graph Q 3. A oempty set S V is a global defesive alliace i G if S is a domiatig set ad δ S (v) δ S (v) 1, v S (1) The global defesive alliace umber of G, deoted by γ d (G), is defied as the miimum cardiality of a global defesive alliace i G. A global defesive alliace of cardiality γ d (G) is called a γ d (G)-set.
A global defesive alliace is called strog if δ S (v) > δ S (v) 1, v S. () or, equivaletly, δ S (v) δ S (v), v S. (3) The global strog defesive alliace umber of G, deoted by γ sd (G), is defied as the miimum cardiality of a global strog defesive alliace i G. A global strog defesive alliace of cardiality γ sd (G) is called a γ sd (G)-set. Cartesia product graphs GivetwographsGadH withsetofverticesv 1 = {v 1,v,...,v 1 }adv = {u 1,u,...,u }, respectively, the Cartesia product of G ad H is the graph G H = (V,E), where V = V 1 V ad two vertices (v i,u j ) ad (v k,u l ) are adjacet i G H if ad oly if v i = v k ad u j u l, or v i v k ad u j = u l. Give a set X V 1 V of vertices of G H, the projectios of X over V 1 ad V are deoted by P G (X) ad P H (X), respectively. Moreover, give a set C V 1 of vertices of G ad a vertex v V, a G(C,v)-cell i G H is the set C v = {(u,v) V : u C}. A v-fiber G v is the copy of G correspodig to the vertex v of H. For every v V ad D V 1 V, let D v be the set of vertices of D belogig to the same v-fiber. Theorem 1. For ay two graphs G ad H of order 1 ad, respectively, we have γ d (G H) mi{ 1 γ d (H), γ d (G)}. Moreover, if G has a efficiet domiatig set, the γ d (G H) γ(g)γ(h). Proof. Let V 1 ad V be the vertex sets of the graphs G ad H, respectively. Let A 1 ad A be global defesive alliaces i G ad H, respectively. We claim that A = A 1 V is a global defesive alliace i G H. It is clear that A is a domiatig set. Now, cosider a vertex (u,v) A. We have the followig: δ A (u,v) = δ A1 (u)+δ H (v) δ A1 (u) 1+δ H (v) = δ A1 (u)+δ H (v) 1 δ A (u,v) 1. Thus, A is a global defesive alliace i G H. Aalogously we prove that V 1 A is a global defesive alliace i G H ad the proof of the upper boud is complete. 3
O the other had, let S = {u 1,...,u γ(g) } be a efficiet domiatig set of G. Let Π = {S 1,S,...,S γ(g) }beavertexpartitioofgsuchthats i = N[u i ]. Let{P 1,P,...,P γ(g) } be a vertex partitio of G H, such that P i = S i V for every i {1,...,γ(G)}. Let A be a γ d (G H)-set. Now, for every i {1,...,γ(G)}, let A i = P H (A P i ). If A i is ot a domiatig set, the there exist w A i such that N Ai (w) =. So, sice S is a efficiet domiatig set, vertex (u i,w) satisfies N Ai (u i,w) = N A (u i,w) =, which is a cotradictio. Thus, A i is a domiatig set i H. Therefore we have that γ(g) γ(g) γ d (G H) = A = A i γ(h) = γ(g)γ(h), i=1 ad the proof of the lower boud is complete. A aalogous procedure gives the followig result for global strog defesive alliaces. Theorem. For ay two graphs G ad H, without isolated vertices, of order 1 ad, respectively, we have γ sd (G H) mi{ 1 γ d (H), γ d (G)}. Moreover, if G has a efficiet domiatig set, the i=1 γ sd (G H) γ(g)γ(h). Proof. By usig the same assumptios tha i Theorem 1 we cosider a vertex (u,v) A ad we have the followig: δ A (u,v) = δ A1 (u)+δ H (v) δ A1 (u) 1+δ H (v) δ A1 (u) 1+1 = δ A (u,v). Thus, A is a global strog defesive alliace i G H ad the upper boud is proved. The lower boud follows from the fact that γ(g)γ(h) γ d (G H) γ sd (G H). Next we improve the upper boud of Theorem 1 by itroducig some restrictios i the graphs used i the product. To do so, we eed to itroduce some termiology. A set S V is a total domiatig set i G if for every vertex v V(G), δ S (v) > 0 (every vertex of G is adjacet to at least oe vertex i S). The total domiatio umber of G, deoted by γ t (G), is the miimum cardiality of a total domiatig set i G. Theorem 3. Let G ad H be two graphs with vertex sets V 1 ad V, respectively. If the order of H is ad, ay two vertices u V 1, v V satisfy δ H (v) δ G (u) 3, the γ d (G H) γ t (G). Moreover, if ay two vertices u V 1, v V satisfy δ H (v) δ G (u), the γ sd (G H) γ t (G). 4
Proof. Let A 1 be a total domiatig set i G. We claim that A = A 1 V is a global defesive alliace i G H. It is clear that A is a domiatig set. Now, cosider a vertex (u,v) A. We have the followig: δ A (u,v) = δ A1 (u)+δ H (v) δ A1 (u)+δ G (u) 3 = δ A1 (u)+δ A1 (u) 3 δ A1 (u) 1 = δ A (u,v) 1. Thus, A is a global defesive alliace i G H ad the boud for γ d (G H) follows. The proof of γ sd (G H) γ t (G) is aalogous to the oe above. Let D 1,...,D k be domiatig sets of graph G with D i = γ(g). For all i deote with G[D i ] the iduced subgraph o vertices D i. We defie the umber Ω(G) = max 1 i k {δ(g[d i])} as the maximum of miimum degrees of all subgraphs of G iduced o ay domiatig set D i, i {1,...,k}, of size γ(g). Theorem 4. Let G ad H be two graphs such that δ(h) (G) Ω(G) 1. The γ d (G H) γ(g) V(H). Proof. Let D = {u 1,...,u m }, m = γ(g), be a miimum domiatig set of G such that Ω(G) = δ(g[d]) ad let V(H) = {v 1,...,v } be the set of vertices of H. The the set S = {(u 1,v 1 ),...,(u 1,v ),(u,v 1 ),...,(u,v ),...,(u m,v 1 ),...(u m,v )}isobviouslyadomiatig set of G H. The set S is also a global defesive alliace sice for every i {1,...,m} ad j {1,...,} it follows that δ S (u i,v j ) = δ D (u i )+δ H (v j ) δ(g[d])+δ(h) = Ω(G)+δ(H) Ω(G)+ (G) Ω(G) 1 = (G) 1 δ S (u i,v j ) 1. The above result give some iterestig cosequeces like the followig oes. Corollary 5. For ay two itegers r,t we have (i) γ d (P r P t ) mi { t r 3,r t 3 }, 5
(ii) γ d (P r C t ) mi { t r 3,r t 3 }, (iii) γ d (C r C t ) mi { t r 3,r t 3 }. Proof. The results follow immediately from Theorem 4, from the fact that, for ay iteger, δ(p ) = 1, (P ) =, Ω(P ) = 0, δ(c ) =, (C ) =, ad Ω(C ) = 0. Thus, if G is a path or a cycle, the the iequality δ(g) (G) Ω(G) 1 is satisfied. The followig lemma together with Theorem 4 leads to some equalities for the global defesive alliace umbers of some Cartesia product graphs which shows that the boud of Theorem 4 is sharp. Lemma 6. [7, 11] For every graph G of order ad maximum degree, γ d (G) ad γ sd (G). +1 +1 +1 The followig result is cosequece of the above lemma ad Theorem 4. Corollary 7. Let G ad H be two graphs beig paths or cycles of orders r ad t, respectively. The { rt t r } γ d (G H) γ sd (G H) mi r,t. 3 3 3 Moreover, if r 0 (mod 3) or t 0 (mod 3), the γ d (G H) = γ sd (G H) = rt 3. Next we cotiue with some other cases of Cartesia product graphs. Propositio 8. For ay two complete graphs K r ad K t we have (i) If r = t, the γ d (K r K t ) = γ sd (K r K t ) = r. (ii) If r t = 1, the γ d (K r K t ) = mi{r,t} ad mi{r,t} γ sd (K r K t ) max{r,t}. (iii) If r t 1, the mi{r,t} γ d (K r K t ) γ sd (K r K t ) max{r,t}. Proof. If r = t, the it is clear that every vertex of oe copy of K r or K t, say K r, has as much eighbors iside the copy as it has outside of the copy ad also, every copy of K r is a domiatig set i K r K t. So, γ d (K r K t ) r ad γ sd (K r K t ) r. Now, sice γ sd (K r K t ) γ d (K r K t ) γ(k r K t ) = r, we obtai (i). If r t = 1, the we ca suppose without loss of geerality that r = t+1 ad let A be the set of vertices of oe copy of K t i K r K t. Notice that A is a domiatig set i K r K t. Hece, for every vertex (u,v) A we have that δ A (u,v) = t 1 = r = δ A (u,v) 1. Thus, A is a global defesive alliace i K r K t ad γ d (K r K t ) mi{r,t}. Now, the equality for γ d (K r K t ) i (ii) follows from the fact that γ d (K r K t ) γ(k r K t ) = mi{r,t} = t. Now, let B be a copy of K r i K r K t. Hece, B is a domiatig set i K r K t ad for every vertex (u,v) B, δ B (u,v) = r = t+1 > t 1δ B (u,v). So, B is a global strog defesive alliace i K r K t ad γ sd (K r K t ) max{r,t}. The lower boud for γ sd (K r K t ) i (ii) follows from the fact that γ sd (K r K t ) γ d (K r K t ) = mi{r,t}. 6
O the other had, suppose t > r +1. Let B be the set of vertices of oe copy of K t i K r K t. Notice that B is a domiatig set i K r K t ad for every vertex (u,v) B we have that δ B (u,v) = t 1 > r 1 = δ B (u,v) > δ B (u,v) 1. Thus, B is a global (strog) defesive alliace i K r K t ad γ d (K r K t ) γ sd (K r K t ) max{r,t}. Fially the lower boud of (iii) follows from γ sd (K r K t ) γ d (K r K t ) γ(k r K t ) = mi{r,t}. 3 Strog product graphs GivetwographsGadH withsetofverticesv 1 = {v 1,v,...,v 1 }adv = {u 1,u,...,u }, respectively, the strog product of G ad H is the graph G H = (V,E), where V = V 1 V ad two vertices (v i,u j ) ad (v k,u l ) are adjacet i G H if ad oly if v i = v k ad u j u l, or v i v k ad u j = u l, or v i v k ad u j u l. Theorem 9. For ay two graphs G ad H of order r ad t, respectively, we have γ d (G H) mi{rγ d (H),tγ d (G)}. Proof. Let V 1 ad V be the vertex set of G ad H, respectively. Let S 1 V 1 be a global defesive alliace i G ad let A = A 1 V. Sice S 1 is a domiatig set i G, we have that A is a domiatig set i G H. Also, for every vertex (u,v) A we have δ A (u,v) = δ A1 (u)+δ(v)+δ(v)δ A1 (u) δ A1 (u) 1+δ(v)+δ(v)(δ A1 (u) 1) = δ A1 (u)+δ(v)δ A1 (u) 1 = δ A (u,v) 1. So, A is a global defesive alliace i G H. Aalogously we prove that V 1 A is also a global defesive alliace i G H, where A is a global defesive alliace i H. Therefore the result follows. Let G be the graph of order six obtaied by joiig with a edge the ceters of two star graphs of order three. Notice that γ d (G) =. Hece, we have that γ d (G K ) = 4 = mi{6 1, }. Thus, the boud of Theorem 9 is tight. Aother case i which this boud is tight is the strog product of two complete graphs K r ad K t of eve orders, where we have that rt = γ d(k rt ) = γ d (K r K t ) = r t = tr. Next, we study some particular cases of strog product graphs. To do so we eed the followig lemma. 7
Lemma 10. [7] For ay iteger, γ d (K ) = +1, γd (C ) = + 4 4, γ d (P ) = + 4 4 if (mod 4) ad γd (P ) = + 4 4 1 if (mod 4). Propositio 11. For ay two itegers r, 4 we have { r +1 ( r r r ) γ d(c r K ) mi r, + 4 }. 4 Moreover, if is a eve umber, the γ d (C r K ) = r. Proof. TheupperboudfollowsdirectlyfromTheorem9adLemma10. LetV 1 = {u 0,u 1,...,u r 1 } be the vertex set of C r (vertices with cosecutive idices are adjacet i C r ad operatios with idices is cosidered modulo r) ad let V = {v 1,v,...,v } be the vertex set of K. Let S be a global defesive alliace of miimum cardiality i C r K ad let (u i,v j ) S. Hece we have that δ S (u i,v j ) δ S (u i,v j ) 1, which is equivalet to δ S (u i,v j ) δ C r K (u i,v j ) 1 = 3. (4) Now, for every i {0,...,r 1}, let S i = S ({u i 1,u i,u i+1 } V ). It is clear that for every i {1,...,}, S i. So, from iequality 4 we obtai that S i 3 +1 = 3. Thus, we have that S 1 r 1 S i 1 r 1 3 3 3 = r i=0 ad the lower boud is proved. Notice that, if is eve, the +1 =. Thus, the upper boud is γ d(c r K ) r ad the equality follows for beig eve. The proof of the ext result is relatively similar to the above proof. Propositio 1. For ay two itegers r, 4 we have (r ) mi { r ( +1, r + r 4 r )} 4, if (mod 4), γ d (P r K ) mi { r ( +1, r + r 4 r )} 4 1, if (mod 4). Proof. TheupperboudsfollowdirectlyfromTheorem9adLemma10. LetV 1 = {u 1,u 1,...,u r } be the vertex set of P r (vertices with cosecutive idices are adjacet i P r ) ad let V = {v 1,v,...,v } be the vertex set of K. Let S be a global defesive alliace of miimum cardiality i P r K ad let (u i,v j ) S. Hece, we have that δ S (u i,v j ) δ S (u i,v j ) 1, ad if i 1 or r, the this is equivalet to i=0 δ S (u i,v j ) δ P r K (u i,v j ) 1 = 3. (5) Now, for every i {1,...,r 1}, let S i = S ({u i 1,u i,u i+1 } V ). It is clear that for every i {1,...,}, S i. So, from iequality 5 we obtai that S i 3 +1 = 3. Thus, we have that S 1 r 1 3 i= S i 1 r 1 3 3 i= = (r ) ad the lower boud is proved. 8
4 Direct product graphs GivetwographsGadH withsetofverticesv 1 = {v 1,v,...,v 1 }adv = {u 1,u,...,u }, respectively, the direct product of G ad H is the graph G H = (V,E), where V = V 1 V ad two vertices (v i,u j ) ad (v k,u l ) are adjacet i G H if ad oly if v i v k ad u j u l. Theorem 13. For ay two graphs G ad H of order 1 ad, respectively, we have γ sd (G H) mi{ 1 γ sd (H), γ sd (G)}. Proof. Let V 1 ad V be the vertex sets of the graphs G ad H, respectively. If A 1 ad A are global strog defesive alliaces i G ad H, respectively, the we claim that A = A 1 V is a global strog defesive alliace i G H. Notice that A is a domiatig set. We cosider a vertex (u,v) A. So, by iequality 3 we have that δ A (u,v) = δ A1 (u)δ(v) δ A1 (u)δ(v) δ A (u,v). Thus, A is a global strog defesive alliace i G H. Aalogously, oe ca proved that V 1 A is a global strog defesive alliace i G H ad the proof is complete. By usig similar techiques like the oes used i Propositios 11 ad 1 for strog product graphs we ca obtai the followig lower bouds. Propositio 14. For ay two itegers r, 4 we have γ sd (C r K ) r 3 + r 3 ad γ sd(p r K ) (r ) 3 + r 3. Proof. Let V 1 = {u 0,u 1,...,u r 1 } be the vertex set of C r (vertices with cosecutive idices are adjacet i C r ad operatios with the subidices are doe modulo r) ad let V = {v 1,v,...,v }bethevertexsetofk. LetS beaglobalstrogdefesivealliaceofmiimum cardiality i C r K ad let (u i,v j ) S. Hece, we have that δ S (u i,v j ) δ C r K (u i,v j ) = =. (6) Now, for every i {0,...,r 1}, let S i = S ({u i 1,u i,u i+1 } V ). It is clear that for every i {1,...,}, S i. So, from iequality 6 we have that S i +1. Thus, we obtai S 1 r 1 S i 1 r 1 (+1) = r 3 3 3 + r 3. i=0 i=0 Now, if V 1 = {u 1,u 1,...,u r } is the vertex set of P r (vertices with cosecutive idices are adjacet i P r ), the by usig a similar procedure we have that S 1 r 1 S i 1 r 1 (+1) = (r ) 3 3 3 i= i= + r 3. 9
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